Algebraic Solutions of Differential Equations 95 To conclude, we simply observe that as c-ar c-br the inequalities in (188.8.131.52) and (184.108.40.206), already established, are neces- sarily strict. This concludes the proof of (6.6.0). (6.6.1) Combining 6.3 and 6.6.0, we are "reduced" to proving the implication (6.2.6)~ (6.2.4) under the additional hypothesis that the rational numbers a, b, c satisfy (220.127.116.11) aCZ, b6Z, c~Z, c-aCZ, c-bc~Z, a-b6Z, c-a-b~Z. (6.6.2) Corollary. In order that rational numbers a, b, c satisfy (6.2.6) and (18.104.22.168), it is necessary and sufficient that for any A eZ which is invertible modulo N for a common denominator N of a, b, c, the rational numbers A a, A b, A c satisfy (6.2.6) and (22.214.171.124). Proof The sufficiency is clear; take A = 1. For the necessity, (126.96.36.199) will still hold because d is invertible mod N, and the condition (188.8.131.52) is obviously invariant under (a, b, c)--* (A a, A b, A c) for such A. By (6.6.0), (184.108.40.206) and (220.127.116.11) imply (6.2.6). (6.6.3) Corollary. In order that rational numbers a, b, c satisfy (6.2.6) and (18.104.22.168), it is necessary and sufficient that for any integers r, s, t, the
This is the end of the preview.
access the rest of the document.